Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.     

Parareal multi-model numerical zoom for parabolic multiscale problems

We present a time-parallel numerical zoom method for parabolic multiscale problems. A fine-finite element solver is defined on a patch where multiscale effects are localized. A coarse finite-element solver, based on a coarser time-space discretization, and possibly smoother than the fine one, is defined on the whole domain of the partial differential equation. The coupling between fine and coarse solvers is carried out with an adaptation of the Parareal algorithm. We detail our numerical method and illustrate it with a numerical experiment.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

Multiscale finite element coarse spaces for the application to linear elasticity

We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169–189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping domain decomposition preconditioners. We motivate and explain the construction and show that the constructed multiscale coarse space contains all the rigid body modes. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our numerical experiments show uniform convergence rates independent of the contrast in Young’s modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness cannot be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589–626]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution.     

Design of a recovery network for WEEE collection: the case of Galicia,Spain

This research aims to optimize the design of the reverse logistic network for the collection of Waste of Electric and Electronic Equipment (WEEE), in the Spanish region of Galicia. As a basis for our study a three-phase hierarchical approach is proposed. In the first phase a facility location problem is formulated and solved by means of a mixed integer linear programming; in the second phase a new integer programming formulation for the corresponding heterogeneous fleet vehicle routing problem is presented, and a savings-based heuristic algorithm is developed to efficiently solve the related collection routing problems; in the third phase a simulation study is performed on the collection routes in order to assess the overall performance of the recovery system. The results show a good performance of the proposed procedure, and an improved configuration of the recovery network compared to the one currently in use (particularly transportation costs are reduced by 29.2%).     

Data compression on the sphere using multiscale radial basis function approximation

We propose two new approaches for efficiently compressing unstructured data defined on the unit sphere. Both approaches are based upon a meshfree multiscale representation of functions on the unit sphere. This multiscale representation employs compactly supported radial basis functions of different scales. The first approach is based on a simple thresholding strategy after the multiscale representation is computed. The second approach employs a dynamical discarding strategy, where small coefficients are already discarded during the computation of the approximate multiscale representation. We analyse the (additional) error which comes with either compression and provide numerical experiments using topographical data of the earth.     

Heterogeneous multiscale methods for stiff ordinary differential equations

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing different time scales. Stability and convergence results for the proposed HMM methods are presented together with numerical tests. The analysis covers some existing methods and the new algorithms that are based on higher-order estimates of the effective force by kernels satisfying certain moment conditions and regularity properties. These new methods have superior computational complexity compared to traditional methods for stiff problems with oscillatory solutions.

     

The heterogeneous multiscale method as a framework for coupling the finite element method and molecular dynamics

Hierarchical two-scale methods are computationally very powerful as there is no direct coupling between the macro- and microscale. Such schemes develop first a microscale model under macroscopic constraints, then the macroscopic constitutive laws are found by averaging over the microscale. The heterogeneous multiscale method (HMM) is a general top-down approach for the design of multiscale algorithms. While this method is mainly used for concurrent coupling schemes in the literature, the proposed methodology also applies to a hierarchical coupling. This contribution discusses a hierarchical two-scale setting based on the heterogeneous multi-scale method for quasi-static problems: the macroscale is treated by continuum mechanics and the finite element method and the microscale is treated by statistical mechanics and molecular dynamics. Our investigation focuses on an optimised coupling of solvers on the macro- and microscale which yields a significant decrease in computational time with no associated loss in accuracy. In particular, the number of time steps used for the molecular dynamics simulation is adjusted at each iteration of the macroscopic solver. A numerical example demonstrates the performance of the model. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

QTT-finite-element approximation for multiscale problems I: model problems in one dimension

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform “virtual” (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ε with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems’ scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem.     

Computational homogenization for heterogeneous cohesive layers incorporating size effects

A numerical framework is presented for the multiscale modelling of material layers that possess a both heterogeneous and microstructured mesostructure. The material layer is represented as a cohesive interface on the macro level, while on the meso level micromorphic representative volume elements are used. A computational homogenization approach for the cohesive material layers has been proposed to solve this nonlinear multiscale problem numerically. The present framework is particularly well suited for the modelling of damage and failure, as the micromorphic RVE is well known for its regularizing character. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rapid Solution of Stokes Flow using Multiscale Galerkin BEM

Two issues that arise when solving Stokes flow problems with the hydrodynamical single‐layer potential are addressed. First, the resulting boundary integral equation is singular, and second, discretizations lead to dense matrices. We discuss a well‐posed modified equation which is equivalent for zero net‐flux. Furthermore, we describe a multiscale basis that lead to sparse stiffness matrices. This approach is suitable for complicated geometries and is an extension of our previous work for the Laplace equation.     

Variational Methods of Micro–Macro Transitions in Nonlinear Elasticity

Continuum micromechanics deals with idealized materials where the macroscopic material response is modelled in an averaged or homogenized sense based on the information of the heterogeneous microstructure. In general, an efficient treatment of multiscale systems requires the application of equivalent structural problems where the constituents are governed by overall properties. The key contribution of this paper is the computational exploitation of variational methods for a numerical upper and lower bound estimation of the effective material response. We present aspects for the formulation of an appropriate minimizing principle yielding the displacement fluctuations on the microstructure and the local effective constitutive variables of the macrostructure depending on the choice whether we apply linear displacement, traction or periodic boundary conditions to the displacement fluctuations on the boundary of the microstructure. The proposed concept will be demonstrated in the scope of some representative model problems.     

Tight frame expansions of multiscale reproducing kernels in Sobolev spaces

Multiscale kernels are a new type of positive definite reproducing kernels in Hilbert spaces. They are constructed by a superposition of shifts and scales of a single refinable function and were introduced in the paper of R. Opfer [Multiscale kernels, Adv. Comput. Math. (2004), in press]. By applying standard reconstruction techniques occurring in radial basis function- or machine learning theory, multiscale kernels can be used to reconstruct multivariate functions from scattered data. The multiscale structure of the kernel allows to represent the approximant on several levels of detail or accuracy. In this paper we prove that multiscale kernels are often reproducing kernels in Sobolev spaces. We use this fact to derive error bounds. The set of functions used for the construction of the multiscale kernel will turn out to be a frame in a Sobolev space of certain smoothness. We will establish that the frame coefficients of approximants can be computed explicitly. In our case there is neither a need to compute the inverse of the frame operator nor is there a need to compute inner products in the Sobolev space. Moreover we will prove that a recursion formula between the frame coefficients of different levels holds. We present a bivariate numerical example illustrating the mutiresolution and data compression effect.     

A computational two scaled model for the simulation of micro-heterogeneous low-alloyed TRIP steels

In transformation induced plasticity (TRIP) steel a diffusionless austenitic-martensitic phase transformation induced by plastic deformation can be observed, resulting in excellent macroscopic properties. In particular low-alloyed TRIP steels, which can be obtained at lower production costs than high-alloyed TRIP steel, combine this mechanism with a heterogeneous arrangement of different phases at the microscale, namely ferrite, bainite, and retained austenite. The macroscopic behavior is governed by a complex interaction of the phases at the micro-level and the inelastic phase transformation from retained austenite to martensite. A reliable model for low-alloyed TRIP steel should therefore account for these microstructural processes to achieve an accurate macroscopic prediction. To enable this, we focus on a multiscale method often referred to as FE² approach, see [6]. In order to obtain a reasonable representative volume element, a three-dimensional statistically similar representative volume element (SSRVE) [1] can be used. Thereby, also computational costs associated with FE² calculations can be significantly reduced at a comparable prediction quality. The material model used here to capture the above mentioned microstructural phase transformation is based on [3] which was proposed for high alloyed TRIP steels, see also e.g. [8]. Computations based on the proposed two-scale approach are presented here for a three dimensional boundary value problem to show the evolution of phase transformation at the microscale and its effects on the macroscopic properties. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy analysis and improved regularity estimates for multiscale deconvolution models of incompressible flows

This paper presents new analytical results and the first numerical results for a recently proposed multiscale deconvolution model (MDM) recently proposed. The model involves a large‐eddy simulation closure that uses a novel deconvolution approach based on the introduction of two distinct filtering length scales. We establish connections between the MDM and two other models, and, on the basis of one of these connections, we establish an improved regularity estimate for MDM solutions. We also prove that the MDM preserves Taylor‐eddy solutions of the Navier–Stokes equations and therefore does not distort this particular vortex structure. Simulations of the MDM are performed to examine the accuracy of the MDM and the effect of the filtering length scales on energy spectra for three‐dimensional homogeneous and isotropic flows. Numerical evidence for all tests clearly indicates that the MDM gives very accurate coarse‐mesh solutions and that this multiscale approach to deconvolution is effective. Copyright © 2015 John Wiley & Sons, Ltd.     

A mixed multiscale finite element method for elliptic problems with oscillating coefficients

The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the large-scale structure of the solutions without resolving all the fine-scale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an over-sampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random log-normal relative permeability to demonstrate the efficiency and accuracy of the proposed method.     

Multiscale method based on discontinuous Galerkin methods for homogenization problems

We propose a multiscale method for elliptic problems with highly oscillating coefficients based on a coupling of macro and micro methods in the framework of the heterogeneous multiscale method. The macro method, defined on a macroscopic triangulation, aims at recovering the effective (homogenized) solution of an unknown macro model. The unspecified data of this model are computed by micro methods on sampling domains during the macro assembly process. In this Note, we show how to construct such a coupling with a discontinuous macro finite element space. We show that the flux information needed in this formulation in order to impose weak interelement continuity can be recovered from the known micro calculations on the sampling domains. A fully discrete analysis is presented. To cite this article: A. Abdulle, C. R. Acad. Sci. Paris, Ser. I 346 (2008).